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Ralf Wittenberg

Department of Mathematics, Simon Fraser University

Research Interests

My general field of research is in dynamical systems, and the following is a brief overview of some major themes; the main emphasis is on analytical and numerical studies of partial differential equations exhibiting spatiotemporally complex and chaotic dynamics, but applications to other fields have also captured my attention...

I welcome inquiries from potential students interested in these or related topics.

Pattern Formation and Spatiotemporal Chaos:

Numerous partial differential equations (PDEs) arising in contexts such as fluid dynamics or surface growth display surprisingly complex temporal dynamics and/or spatial pattern formation.  The Kuramoto-Sivashinsky (KS) equation is a particularly rich (and much-studied) example, and I have long been interested in investigating various aspects of spatiotemporal chaos in the KS equation and its generalizations, analytically and numerically.

In recent years I have focussed especially on a related 6th-order PDE, the Nikolaevskiy model for short-wave pattern formation with Galilean invariance, and its associated Matthews-Cox modulation equations.  Work with my former student Philip Poon revealed spatiotemporal chaos with strong scale separation, potential anomalous scaling, Burgers-like viscous shocks and coarsening phenomena to chaos-stabilized fronts; there is much that remains to be done to understand this curious dynamical behaviour!

Time evolution of Kuramoto-Sivashinsky equation    Time evolution of Nikolaevskiy equation

Applied Analysis:

While the solutions of such nonlinear PDEs are typically too complex to permit detailed analytical description, rigorous functional-analytic estimates on global, long-time or averaged properties of solutions on the attractor may nevertheless often be proved. Viscous shock in destabilized KS equationI am especially interested in the interplay between numerical and analytical results; as an example, my numerical discovery and asymptotic investigation of a viscous shock solution in the destabilized KS equation influenced subsequent improvements in, and constraints on, rigorous bounds on the scaling of the absorbing ball for the KS equation.

A related major theme of my research concerns analytical estimates in fluid dynamics, notably turbulent Rayleigh-Bénard convection, for which I am particularly interested in establishing rigorous a priori variational bounds on averaged quantities such as bulk convective heat transport.


I have collaborated on and (co-)supervised students interested in dynamical models in various areas, including mathematical epidemiology and immunology, aggregation models and opinion dynamics.  For much of this research, I am associated with the IMPACT-HIV group (based at the IRMACS Centre at SFU), an interdisciplinary research team studying differential equation and network models of the HIV epidemic, with a particular focus on evaluating Treatment as Prevention control strategies.